The shape of space-time around a star

Gravity around star is often depicted as the curvature if spacetime or a 'dimple" in a flat sheet.

What would happen to space-time around a star if the star were to instantly disapear? Would the fabric of space time snap back to flat space or would it reverberate, expanding then contracting then expanding again?

The space - time around a spherically symmetric, static star (which many stars are to a good approximation) has similar properties to a 4 - sphere; if you looked at a constant radius, constant time section of the geometry then it will be equivalent to that of the 2 - sphere. It isn't a flat sheet by any means, that analogy hurts more than it helps. Your question, unfortunately, requires extrapolation of a situation that is physically impossible. You can't just make a star disappear out of nowhere even in principle.

Staff: Mentor

Hi RayYates, welcome to PF!

As WannabeNewton says the problem is that the star simply disappearing is a violation of the Einstein Field Equations. And those are the very equations which we would use to determine the curvature of the spacetime.

However, you could imagine that the star suddenly turns into a spherically symmetric ball of light going outwards. The shape of the spacetime outside of the ball of light would be the same as before, but I don't know what the shape would be inside. I think that it would be flat.

Staff: Mentor

Well, I think something similar to a gravitational wave would be released. We cannot remove a star instantly, but we can imagine a high mass object moving at near c velocity past us. It would move through space and the curvature would follow it with the resulting changes propagating through space at c as a wave. That is about as close to a massive object "dissapearing" that I can imagine.

Even in a thought experiment you cannot just violate physical laws. The notion of positive and negative curvature does exist as an intrinsic measure for manifolds and this property is called Gaussian curvature. But again, your scenario just isn't possible even in principle.

There is nothing in the EFEs that prevents that. In fact, the Friedman metric with negative curvature index describes hyperbolic space, at each instant of time, which has a negative Gaussian curvature.

If mass can cause space time to curve toward the mass, (lets call that a positive curvature of space time) are there any reason space time can not have a negative curvature?

I think you have a misunderstanding of the terminology. Positive curvature means curvature like a spherical surface. Negative curvature means curvature like a saddle.

I think you mean something different, essentially "concave" and "convex" curvature. Well, there is no difference. You have to remember that when you picture 4-dimensional spacetime curving in 5-dimensional space, the 5th dimension doesn't really exist, it's just an analogy to help you picture the geometry.

Note that it's important to realize that you get nonsense, if you don't realize this, you can spend a lot of non-productive time arising from false initial assumptions.

I understand the two dimensional representation of a ball on a "flat" sheet as curved space time is only poor representation of gravity.

I'm trying to understand, if mass can cause space time to curve toward the mass, (lets call that a positive curvature of space time) can space time have a negative curvature?

Positive and negative curvatures exist, but they don't have anything to do with whether the sheet goes up or down. The shape is the same.

Will the sheet bounce / reverberate if the ball were suddenly removed?

As was mentioned previously, the only honest answer to that question is "that can't happen". However, other sorts of disturbances can cause gravity waves, which you can think of as ripples in space-time, or in your analogy ripples passing along the sheet.

Yes a super - nova could generate gravitational waves (note that gravity waves are different). If the collapse of the star happened to be spherical then no gravitational waves would be produced but if there was a non - spherical collapse then the non - spherical parts will be radiated away as gravitational waves. The collapse of a star is a complex process so the amplitude of the gravitational waves could probably only be found numerically (too many self - interactions to use the non - linear EFEs to find a solution for the gravitational waves with a collapsing star as the source term) so I can't do it for you with a pen and paper but try googling it, there might be published papers on the amplitude of gravitational waves released from non - spherical collapse of stars (or maybe an adviser/mentor already has a resource ready).

Interesting point! Behaving similarly to as if it were refracting through a layer of something with higher optical density, perhaps? So the ultimate effect of the gravity wave (if they do in fact affect light in this manner) would be to leave the light traveling in the same direction, but its course would be shifted slightly sideways.